Methodology of the linear solver Governing equations

In cylindrical coordinates and by assuming harmonic variation of the form exp(i ш t) and exp(im) in time and in the azimuthal direction, respectively, the linearized Euler equations for 1D or axisymmetric configuration can be described by:

du du 1 dp duo duo p dpo

гит + u0——b v0——I————- ——Ь и— Ь v——————————- — —

dx dr po dx dx dr po Ypo dx

dv dv 1 dp dvo dvo p dpo

iojv + u0 ——b v0——I ——b u———– Ь v———————— —

dx dr po dr dx dr po Ypo dr

dw dw im vo w

iujw + u0— + v0— b p b

dx dr po y r

Equations 1, 1, 3 and 4 are solved on the curvilinear mesh using uniform computational coordinates (£, n). Metrics terms and spatial derivatives are computed using the optimised 4th order DRP scheme from Tam & al. [Tam and Webb, 1993]. The numerical scheme is stabilized for the spurious wave by adding a selective damping term for cartesian grid or by filtering the so­lution [11] for curvilinear grid. To solve equations 1, 1, 3 and 4, a pseudo­time derivative is introduced so that the equations may be marched to a steady state condition using conventional computational fluid dynamic scheme. In

the present investigation, the pseudo-time technique is discretized using a sim­ple forward-Euler scheme. At each iteration, the selective damping and/or the filter (8th order) are applied as well as the boundary condition.